Since all terms in equations (REF ) and (REF ) are proportional to either \(({\nabla } \overline{T})^2\) , or \(({\nabla } \overline{\rho })^2\) , or \(({\nabla } \overline{T}) ({\nabla } \overline{\rho })\)  (see below), and we consider homogeneous
density stratified turbulence, we do not need to perform additional Taylor expansions
over small parameters \(\ell _0/H_T\)  and \(\ell _0/H_\rho \)  in these
terms [1]}, [2]}, where \(H_T\)  and \(H_\rho \)  are the characteristic scales of variations of the mean temperature and mean density, respectively. In particular, hereafter we neglect small terms \(\sim {\rm O}[(\ell _0/H_T)^3, (\ell _0/H_\rho )^3]\) .
This implies that we replace \({k}_1\)  by \({k}\)  and \({k}_2\) 
by \(-{k}\)  in all second moments in equation (REF ).
